The difference between the filters you name is not that each new one invented made a closer approximation to the ideal filter, but that each one optimizes the filter for a different characteristic. Because there's a trade-off between different characteristics, each one chooses a different way to make this trade-off.
Like Andy said, the Butterworth filter has maximal flatness in the passband. And the Chebychev filter has the fastest roll-off between the passband and stop-band, at the cost of ripple in the passband.
The Elliptic filter (Cauer filter) parameterizes the balance between pass-band and stop-band ripple, with the fastest possible roll-off given the chosen ripple characteristics.
Now if I was to take my 5th order structure and was able to simulate for every possible inductor value and capacitor value would I find a combination that would give me the best possible / closest model to ideal, that beats all previously known filter types?
It depends what you mean by "best possible" or "closest model". If you mean the one with the flattest response in the pass-band, you'd end up with the Butterworth filter. If you mean the best possible roll-off given a fixed ripple in the pass-band, you'd end up with the Chebychev design, etc.
If you chose some other criterion to optimize (like mean-square error between the filter characteristic and the boxcar ideal, for example), you could end up with a different design.
Do mathematicians / engineers know of a "best" filter response that is physically possible for a given order but so far do not know how to create it.
The filters you named (Butterworth, Chebychev, Cauer) are the best, for the different definitions of "best" that define those filters.
If you had some other definition of "best" in mind, you could certainly design a filter to optimize that, with existing technology. Andy's answer names a couple of other criteria and the filters that optimize them, for example.
Let me add one other question you might ask as a follow up,
Why don't we in practice design filters to optimize the mean-square error between the filter characteristic and the boxcar ideal?
Probably because the mean-square error doesn't capture well the design-impact of
"errors" in the pass-band and stop-band response. Because the ideal response has 0 magnitude in the stop-band it's hard to define a "relative response" measurement that has equal weight in both regions.
For example, in some designs an error of -40 dB (.01 V/V) relative to the ideal 0 V/V response in the stop-band would be much worse than an error of 0.01 V/V in the passband.
The first thing to do when looking to design a filter for a signal is to obtain the spectrum of the signal. This can be done in different ways. For example, you can measure or capture the signal with a scope. Most modern scopes can calculate an FFT of captured data and give a spectrum.
For a design though, it is nice to make an idealized waveform and then write the Fourier series of it. This way you get only the signal and no unwanted stuff, and so get a clean spectrum which will just be the coefficients of the harmonics. Then you can truncate or alter harmonic content (coefficient amplitudes) according to some filter type. Reconstruct a filtered signal from those altered coefficients and the filter effects will be evident.
From the comments of having a 5MHz carrier with ~1MHz BPSK, it is likely that many harmonics will be needed if a good representation of the original signal is desired. It wouldn't be surprising if 20 to 40 harmonics were needed for good representation. That would be a filter that started to roll off at > 20MHz, or maybe 40MHz. That's only 2 or 3 octaves from the 2 meter band (~150MHz). A filter with 1st order roll off isn't going to do that. You're going to need a filter with at least 3rd order performance for something like that. It's possible to get 2nd order performance out of a single stage Sallen-Key low pass filter.
As to the design shown in the schematic, it looks like the amplifiers are OK for what's being asked of them. LC filters are sensitive to termination. Best performance is when they are terminated into their characteristic impedance. The LC shown has \$Z_o\$ of 45 Ohms, and when used alone shows a resonant lobe of about 6dB. You can see the effects of impedance matching in case 4 when the LC is terminated into U2 with 100 Ohms. For a better match, R8 and R7 values could be halved, or L1 could be doubled and C5 could be halved.
Usually when active filters are used, inductors are not. It's because inductors are not needed to get complex poles and high Q roll offs, amplifier gain (and sometimes a little positive feedback, as in Sallen-Key) give that. Passive filters are good if the filter will be someplace where there is no bias voltage, or where frequencies and or filter order is high.
Best Answer
A low pass filter is as follows:
$$ \frac{Vout}{Vin}= \frac{1}{\tau*s+1} $$
where \$\tau\$ is equal to \$ RC\$.
Since they are linear in the frequency space they multiply:
$$ \frac{Vout}{Vin}= \frac{1}{\tau_1*s+1}*\frac{1}{\tau_2*s+1}*\frac{1}{\tau_3*s+1}*\frac{1}{\tau_4*s+1}*\frac{1}{\tau_5*s+1}\ .$$
If your tau's are all the same then it would be: $$ \frac{Vout}{Vin}= \left( \frac{1}{\tau*s+1} \right)^5\ .$$
Realizing these filters are different, as each RC stage will present a load to the stages after that, which is why we use op amps to isolate the impedance from each stage.